Singular polarizations and ellipsoid packings.
Abstract
We prove in this paper that any dimensional symplectic manifold is essentially made of finitely many symplectic ellipsoids. The key tool is a singular analogue of Donaldson’s symplectic hypersurfaces in irrational symplectic manifolds.
1 Introduction.
Donaldson proved in [6] that a symplectic manifold with (socalled rational) always admits a symplectic polarization of large enough degree , that is a symplectic hypersurface Poincarédual to . In [5], Biran showed that these polarizations decompose the manifold into a standard ”fat” part and a ”thin” part which is isotropic in the Kahler case, and which has zerovolume in any case. In [14], it was noticed that the standard part of the previous decomposition is itself made of a standard ellipsoid and an object of codimension one. Put together, these results show that rational symplectic manifolds are always covered by one ellipsoidal Darboux chart up to a negligible set. This approach is rather satisfactory for or where polarizations of low degrees can easily be found. However, as the degree of the polarization becomes larger, the ellipsoid gets more intricate and the codimensionone part more significant. This explosion of degree prevents from getting anything interesting on irrational manifolds. The present work shows however that an analogous result holds in the irrational setting.
Theorem 1.
Any closed dimensional symplectic manifold has full packing by a finite number of ellipsoids. This number can be bounded by a purely topological quantity : the dimension of .
This theorem is not really about symplectic embeddings : it does not address the question of how flexible they might be, like for instance [17, 13, 10, 8, 9]. It rather gives a description of a symplectic manifold as a patchwork of euclidean pieces (ellipsoids) whose complexity  if only measured by the number of pieces  does not really depend on the symplectic structure (see also [16] for a result in this spirit). Although the bound above is rather loose (for instance when the symplectic form is rational), it can be improved by a closer look at the proof. In fact,
The theorem is a consequence of the following two results. First, Donaldson’s construction of polarizations extends to irrational symplectic manifolds.
Theorem 2.
For any symplectic manifold there exist symplectic hypersurfaces with transverse and positive intersections such that
(1) 
We can also assume that the classes are independent in .
A family of symplectic hypersurfaces that satisfies (1) will be called a singular polarization of . In dimension higher than four, the meaning of ”positive intersection” obviously has to be explained, and we refer the reader to section 5. As their classical analogues, singular polarizations can be used to embed ellipsoids, at least in dimension four.
Theorem 3.
Let be a closed symplectic manifold with
where are symplectic curves whose pairwise intersections are all positive and whose Poincaréduals are independant in . Then has a full packing by the ellipsoids where denotes the symplectic area of . Precisely, for all , there exists an embedding
which admits as supporting surfaces, i.e. the image of the ”horizontal” disc in covers up to an area .
Some remarks are in order. First, a simple computation shows that is covered by the image of up to a volume of order , hence the wording ”full packing”. Together with theorem 2, it obviously proves the basic assertion of this paper. Next, in theorem 3, the curves are allowed to have negative selfintersections : the positivity condition only concerns intersections between different curves. As such, it applies for instance naturally in the context of blowsup. It can therefore be used to understand what happens to the ellipsoid decomposition in rational sympletic manifolds equipped with polarizations with singularities. It allows in some sense to make the desingularization process compatible with Biran’s decompositions. Another application concerns symplectic isotopies : the proof of theorem 3 goes along the same lines as the proof of Biran’s decomposition theorem given in [15], and it extends the range of the method of isotopy developed there. Finally, it may be worth pointing that both the dimension hypothesis and the independance of the Poincaréduals seem mostly technical, and can be removed at least in some concrete situations (e.g. with a polarization consisting of two linear hyperplanes is good enough).
The paper is organized as follows. We first discuss the main idea of the paper through the two easiest examples : the nonsingular and the ”flat” cases. In section 3, we give a local model for a neighbourhood of a singular polarization, as well as the main properties of this model in terms of Liouville forms. In section 4, we prove theorem 3. We then explain the small modifications to Donaldson’s arguments needed to prove the existence of singular polarizations (theorem 2). We finally deal with the applications in the last two sections : Biran’s decomposition associated to singular curves in section 6 and isotopies of balls in section 7.
Notations :
We adopt the following (not so conventional) conventions throughout this paper :

All angles will take value in . In other terms an angle is a full turn in the plane, and the integral of the form over a circle around the origin in the plane is .

The standard symplectic form on is , where are polar coordinates on the plane factors. With this convention, the euclidean ball of radius has capacity .

A Liouville form of a symplectic structure is a oneform satisfying . The standard Liouville form on the plane is .

A symplectic ball or ellipsoid is the image of an euclidean ball or ellipsoid in by a symplectic embedding.

The Hopf discs of an euclidean ball in are its intersections with complex lines.

denotes the dimensional ellipsoid . Because of our normalizations, its Gromov’s width is .
2 Two easy examples.
2.1 The nonsingular case.
In this paragraph we review briefly for selfcontainedness the result of [14] in the setting of smooth polarizations. Let be a rational symplectic manifold with a polarization of degree . Biran’s result states that there is an embedding of a symplectic disc bundle into which has full volume. This disc bundle can be seen as the part of the normal line bundle of  denoted by  in on which the closed form (to be defined soon) is symplectic. The line bundle can be equiped with a hermitian metric and a connection form which allow to define a form on satisfying and . The form is then simply given in these coordinates by :
It was proved in [14] that the restriction of this disc bundle to a disc of area in the base is an ellipsoid .
Lemma 2.1.
Let be the symplectic disc bundle defined above and let be a disc of area in . Then .
Let us mention an application that was not made completely explicit in [14]. It answers a question of McDuff [10].
Theorem 4.
There exists a symplectic embedding of into .
Proof : First notice that has such a full packing because it has a polarization of degree , of area ,
namely a conic. Let us give now an explicit description of a prefered disc bundle over the conic . Since is real, it is invariant by conjugacy, and each real projective line intersects
in exactly two distinct conjugated points. Moreover,
splits all these lines into two disks of equal area onehalf, that contain one of these two points each. The fibers of the disc bundle over the points of are precisely these half real lines [5].
Fix now and call the (real) line passing through and . Consider also a disc of full
area which misses and . The restriction of this symplectic disc bundle to is an open ellipsoid . By construction, this ellipsoid does not meet the fibers above , so it misses
the projective line . Since , the ellipsoid embeds in fact into .
Lemma 2.1 serves also to split an ellipsoid into smaller ones. As such, it proved useful to give a natural construction of a maximal symplectic packing of by five balls [14]. Let us now mention a far less successful story : looking for such a maximal symplectic packings of by seven balls (known by [11] to be of radius ). Using the same idea, one can easily pack with eight ellipsoids using a smooth polarization of degree three. These ellipsoids fail to contain the desired balls because . But there are eight of them instead of seven. Notice that one of these ellipsoids can then be split into eight ellipsoids . In this approach, the question would now to be able to glue seven of these eight thin ellipsoids with the seven bigger ones to get seven ellipsoids . But this points seems rather hard.
2.2 The product case.
Let us discuss now the basic idea of the paper, in the easiest case of a product. Consider the symplectic manifold , where are relatively prime integer. This manifold has a symplectic polarization of degree which is a smoothening of a curve
where are selfmaps of of degrees and respectively. Over this complicated polarization, there is a symplectic ellipsoid which cannot be very simple. For instance, when degenerates to an irrational number, Gromov’s capacity of the ellipsoid collapses, and nothing remains at the limit. By contrast, there is a much simpler singular polarization on the homological level given by , which provides a decomposition of into two ellipsoids in the following way.
Put coordinates on (remember that ) with the convention that and is one point ( is seen as the one point compactification of the disc of suitable radius). Denote also and . The symplectic form on is
The Liouville form is defined on and gives rise to a forward complete Liouville vector field, which is easily seen to be
The action of this vector field is best seen on the toric coordinates on , and is shown in figure 1.
We actually see that is tangent to the line , so the trajectories of emanating from and are respectively and . These triangles are wellknown to be filled by the ellipsoid . Thus we see that we get the toric decomposition of into two ellipsoids (up to zero volume) out of a data consisting of a singular polarization and a Liouville vector field on the complement of . This approach provides much simpler objects (in a geometric sense) than the one giving only one ellipsoid. In particular, both the singular polarization and the embeddings survive the process of degenerating to an irrational. The aim of this note is to understand this simple picture in a general context.
3 Plumbed symplectic disc bundles.
Let be as in theorem 3, that is the are symplectic smooth curves with
and all intersection points between any two of these curves is positive. Put . With no loss of generality, we can assume that the curves are symplectic orthogonal with respect to at each intersection point (such a configuration can be achieved by small local perturbations).
3.1 Local model near the polarization.
Decompose first the area form on as , where :

the forms have supports on small discs around , with total masses ,

the form has support on the complement of smaller discs also centered on , with total mass
We can also assume that the area of on the complement of the discs is for slightly smaller than .
Consider now the line bundle which is modeled on the (symplectic) normal bundle of in  i.e. they have the same Chern class. Endow this bundle with a hermitian metric, (local) coordinates and a connection with curvature , where
Notice that is negative when and vanishes when . Defining the form on by asking that its restriction to the fiber is and that it vanishes on the horizontal planes of the connection, we get a form that checks :
We define now a closed twoform on by
When is nonpositive, this form is symplectic on . But in the positive situation, is only symplectic on the disc bundle of area (on even larger discs over ). We will denote in the sequel by the symplectic part of the line bundle.
A standard Moser argument shows moreover that there are some neighbourhoods of the zerosection and respectively which are symplectomorphic. In other terms, there exists an embedding
For simplicity, we henceforth assume that is itself endowed with a fibration (given by ) and coordinates . Moreover, since and are symplectic orthogonal at , a local symplectomorphism allows to make the fibration structures of and coincide in , namely arranging that provide full coordinate charts in , for which the two set of fibres are given by the fibres of and . With such normalization, we can finally assume that
(2) 
where outside and coincides with near . In some neighbourhood of this point, we therefore have :
Let us sum up the above discussion:
Proposition 3.1 (Weinstein).
There exist neighbourhoods of in and of the zerosection in which are identified via diffeomorphisms . The expression of the symplectic form in these coordinates is given by
where has support in and
Finally, near , .
Otherwise stated, a neighbourhood of the whole polarization is a plumbing of the along the bidiscs (where ).
3.2 Liouville forms on the symplectic disc bundles.
The symplectic disc bundles defined in the previous paragraph come naturally with Liouville forms (recall they are primitives of the opposite of the symplectic forms). A more careful analysis  that we perform now  shows that it is possible to impose compatibility conditions on these forms, which allow to glue them to get a Liouville form on .
Lemma 3.2.
There is a Liouville form on such that near . In fact,
(3) 
for wellchosen Liouville forms , for , in . The Liouville form can however be chosen arbitrarily on any disc compactly supported in .
Proof : Consider first any Liouville forms for in . Then the oneform defined by (3) is a Liouville form for . Indeed,
We now need to choose well the forms and . Define first by
and recall that by definition of , it vanishes identically outside . In order to define , notice that has support in and . Therefore, there exists a Liouville form of such that
It is moreover obvious that this condition is compatible with any requirement on on a disc compactly supported in . Putting all this together, we get the following expression for in the neighbourhood of :
Recall that a Liouville form gives rise to a vector field  called Liouville  by symplectic duality : . This vector field has the property of contracting the symplectic form : . Thanks to the cautious choices we made until now, both the sets of Liouville forms and vector fields glue together to welldefined objects on .
Lemma 3.3.
The formulas
define a Liouville form and its associated Liouville vector fields on . Moreover, the vector field points outside if this neighbourhood is wellchosen.
Proof : The first point is an obvious consequence of the previous lemma because near . The second statement is a straightforward consequence from the fact that each points outside the zerosection on , and this is a simple computation :
Thus near the zerosection .
The following lemma gives a nice expression of the Liouville vector fields associated to the forms defined above. In the statement, the disc should be thought of as a disc of of approximately full area.
Lemma 3.4.
Consider the trivial disc bundle (or if ) over a disc in , with polar coordinates and on and respectively. Equip this bundle with the symplectic structure , where and . Let be a Liouville form for defined by
and its associated vector field. Then


there exists a smooth function such that the map
is a symplectomorphism (when is negative, is an hyperboloid rather than an ellipsoid);

setting and ,
Proof : The point ii) is word for word the same statement and same proof than lemma 2.1 in [14]. It is an easy computation, which we do not repeat here. The point i) is a simple verification. Write and compute :
For iii), express first in the good coordinates :
(4) 
Then,
(5) 
Taking (4) and (5) into account, we therefore get :
3.3 Ellipsoids in the standard bundles.
The ellipsoids of theorem 3 naturally arise from as the set of points that can be reached by flowing out of a disc in along the Liouville vector field. Precisely,
Proposition 3.5.
Let be a disc of symplectic area in , viewed as the zerosection of . Then, if the form is wellchoosen on , the basin of attraction of this disc, defined as
is symplectomorphic to the ellipsoid .
Proof : Since is contained in , the symplectic form on the restriction of to is exactly of the form of lemma 3.4 :
Provided corresponds also to the Liouville form called ”standard” in this lemma (which can always be achieved because can be any Liouville form on by lemma 3.2), it provides a symplectic embedding . This map sends the set to
By lemma 3.4 iii), if are coordinates on , and , the differential equation associated to is
Now is the set of points that verify :
() 
An easy computation shows that , so that () writes . This in turn means
We conclude this paragraph by noting that this ellipsoid is contained in the part of the bundle above the disc simply because of the formula i) of lemma 3.4. Indeed, since , the ”horizontal” part
of the vector field above points inside .
Remark 3.6.
The set lies inside .
3.4 Variations of the Liouville forms.
Liouville forms are never unique : they can always be modified by adding a closed oneform. In the previous paragraphs, we needed to impose several compatibility conditions for the Liouville forms, namely fix them on discs . These requirements only rigidify slightly the situation but still leaves a lot of freedom, which will be fully needed in the proof of theorem 3. Precisely, we will need the following set of objects :

A family of closed oneforms on which vanish identically on all the and . Notice that all homological classes in have such representatives.

A family of Liouville forms on .
These forms obviously satisfy the same compatibility conditions as the , i.e. they give rise to a welldefined Liouville form still denoted on . Moreover, since in (and therefore in ), the remark 3.6 ensures that proposition 3.5 holds when is replaced by . Finally, since differs from only by a pullback by , the radial component of its Liouville vector field does not change : it still moves away from the zerosection, so that lemma 3.3 also holds for .
4 Proof of theorem 3.
We adopt in this paragraph all conventions, notations and results of section 3 . The core lemma is now the following :
Lemma 4.1.
There exists a family of oneforms on which vanish identically on and such that the form defined on extends to a Liouville form on .
Let us first explain quickly why theorem 3 is a direct consequence of this lemma. Since is compact and points outside the , it defines a forwardcomplete vector field on . Therefore, and since is really an extension of , the elementary dynamical procedure that consists in extending the local symplectic embeddings (given by prop 3.1) by
provides symplectic embeddings which overlap, but clearly not on the sets . We therefore have an embedding
which is the desired ellipsoid packing by proposition 3.5.
Proof of lemma 4.1 : First observe that by definition of the curves , the symplectic form vanishes on any cycle of , so it is exact on , and we can pick a Liouville form for on this set. In , the difference is closed. If it is moreover exact, the lemma follows because any extension of the function defined by gives an extension of to the whole of . We explain now that although this difference may well not be exact, we can find a ”correction” closed oneform as in paragraph 3.4 such that is exact. To understand this point, consider a family for the onedimensional homology of , where is the small loop around (contained in a fibre of and defined by the equation ) and the are lifts of simple closed loops in which span .
We first prove that vanishes for all . Since tends to when goes to zero, also has a limit, . Consider now a twocycle and perturb it so that it becomes transverse to the curves . Then since , we have :
Thus, vanishes in which implies the vanishing of each by the independence hypothesis. Notice that we only use this hypothesis at this point, so when the are contractible for instance, the independance is not needed.
Define now by requiring that . Provided that we were cautious to take with no intersection with and , we can even require to vanish on these discs. Then a simple computation (explicitly made in [15]) shows that vanishes on each class . Moreover, since , its values on the loops remain unchanged, so that